# A comparison principle for SDE

Let $$W$$ be a standard one dimensional Brownian motion, and $$\mathcal F_t$$ its natural filtration. Consider the SDE

$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$

$$dY_t = \mu_Y (t, \omega) \, dt + \sigma_Y (t, \omega) \, dW_t$$

$$X_0 = x_0, Y_0 = y_0 \text{ a.s.}$$

where $$\mu_X, \mu_Y, \sigma_X, \sigma_Y \geq 0$$ are progressively measurable with respect to $$\mathcal F_t$$, and $$x_0, y_0$$ are constants.

Assume the existence of a solution to the above two equations up to a determinstic time $$T$$.

Question: Suppose $$\sigma_X \neq \sigma_Y$$ on a subset of $$\Omega \times [0, T]$$ of positive measure. Then is it true that

$$\mathbb P(Y_T > X_T) > 0?$$

Don't think so. Take both to satisfy something like $$dZ = \sigma |Z| dW$$, start one from -1, and one from 1. regardless of the exact parameters, one stays positive and one stays negative.